The capacity of (terrestrial or satellite) mobile-radio telecommunications systems is limited by co-channel interference. Specifically, although by virtue of digital beamforming it is possible for a plurality of remote users to share the same frequency resource (the same channel), the spatial isolation required for this sharing is however not perfect and inevitably leads to the appearance of interfering signals. Complete or partial suppression of these interfering signals is therefore a key objective of the digital signal processing associated with beam formation.
In the case of digital beamforming in reception (FIG. 1), it is necessary to consider N radiating elements of an antenna array receiving signals x(t)=[x1(t), . . . , xN(t)]T that are converted into digital samples x[k]=[x1[k], x2[k], . . . , xN[k]]T, N being an integer higher than or equal to 3. The digital samples allow a correlation matrix Rx containing information on the arrival directions of the signals, which consist of a useful signal, interfering signals and noise, to be constructed. The interfering signals are also referred to as interference.
The vector x[k] of the samples of the signal received by the N radiating elements of the antenna may be written:x[k]=[x1[k],x2[k], . . . ,xN[k]]T=su[k]+si[k]+sb[k]  (1)where x[k] represents the K vectors of dimension N×1 of the samples of the received signals, k being a natural integer comprised between 1 and K, K being the number of samples, su[k] being the useful signal and si[k]+sb[k] the interference and noise.
The narrowband assumption is assumed to apply to the array (H. L. Van Trees, “Optimum Array Processing”, Part IV of Detection, Estimation and Modulation Theory, Wiley Interscience, 2002), i.e. the maximum propagation delay between the radiating elements of the antenna is assumed to be very much shorter than 1/(πB), where B is the bandwidth of the signal. In this case, the vector of the useful signal su[k] may be written su[k]=u[k]a(θu, ϕu), with u[k] the amplitude of the useful signal and a(θu, ϕu) the directional vector of the antenna in the direction (θu, ϕu) of the useful signal. Under the narrowband assumption, the directional vector is independent of the frequency of the signal.
Likewise, the interference vector si[k] may be expressed as follows: sis[k]=Σl=1Lil[k]a(θil, ϕil), with il[k] the signal of the l-ith interfering signal and a(θil, ϕil) the directional vector of the antenna in the direction (θil, ϕil) of the l-ith interfering signal.
The directional vector a(θ, ϕ) is a vector in the algebraic sense, and more precisely a vector of complex functions of the two variables θ and ϕ. It represents the amplitude and phase response of the N radiating elements of the antenna array in the direction (θ, ϕ).
This directional vector may be measured in the laboratory for a given antenna and for a set of directions (θ, ϕ). A numerical or analytical model of the directional vector may also be available. For example, for an equispaced linear array (ELA) of N radiating elements, the n-th component of the vector a(θ, ϕ) may be written (with ϕ=90°):
                                          a            n                    ⁡                      (            θ            )                          =                  exp          (                      j            ⁢                                          2                ⁢                π                            λ                        ⁢                          d              ⁡                              (                                  n                  -                  1                                )                                      ⁢            sin            ⁢                                                  ⁢            θ                    )                                    (        2        )            where λ is the wavelength and d the distance between the radiating elements.
Nevertheless, when the antenna is in operation, it may be subjected to temperature variations or geometric deformations. The radio-frequency reception chains located after the antenna are also subjected to temperature drifts and are not completely identical. As a result, the “current” directional vector (that of the antenna in operation) is different from the “assumed” directional vector (that measured in the laboratory or given by a model). The difference between these two vectors, for a given direction, is a vector of errors commonly referred to as “amplitude/phase errors”, since they affect both the amplitude and the phase of each component of the directional vector. These amplitude/phase errors have a negative effect on the performance of most interference suppression methods. These errors may be dependent on the direction of the signal, such as in the case of geometric deformations for example, or independent of direction, as in the case of disparities between reception chains.
The signal y[k] of the formed beam may be written:
                              y          ⁡                      [            k            ]                          =                                            ∑                              n                =                1                            N                        ⁢                                          w                n                *                            ⁢                                                x                  n                                ⁡                                  [                  k                  ]                                                              =                                                    w                _                            H                        ⁢                                          x                _                            ⁡                              [                k                ]                                                                        (        3        )            where wH is the conjugate transpose of the complex weighting vector w=[w1, . . . , wN]T.
The formed beams, such as shown in FIG. 1, may therefore be written: y1[k]=w1Hx1[k] for beam 1, y2[k]=w2Hx2[k] for beam 2 and ym[k]=wMHxM[k] for beam M.
To form a beam directed toward the useful signal and suppress interference, it is necessary to compute a complex weighting vector such as to make the signal output from the beam as similar as possible to the useful signal, i.e. such that y[k]≈u[k]. This is equivalent to saying the signal-to-noise-plus-interference ratio (SINR) must be maximized. This ratio is an indicator of the quality of the communication and is directly related to the maximum throughput of the link.
The correlation matrix Rx of the received signals, representing the interdependence of the samples of the signal received by the N radiating elements of the antenna, is here written Rx=Ru+Rib, with Ru the correlation matrix of the useful signal and Rib the correlation matrix of the interference and noise.
The estimate of the matrix Rx computed from the samples x[k] of the received signal vector is the basic information that is used by all adaptive interference suppression methods.
The SINR may be expressed directly as a function of the correlation matrices and of the weighting vector w to be computed:
                    SINR        =                                                            w                _                            H                        ⁢                          R              u                        ⁢                          w              _                                                                          w                _                            H                        ⁢                          R              ib                        ⁢                          w              _                                                          (        4        )            
The aim of interference suppression methods is to maximize SINR and they often implement an inversion or diagonalization of the correlation matrix Rx.
Depending on the system, there are three conventional situations in which interference suppression methods are implemented:                1st situation: The emission position of the useful signal, also called the arrival direction of the useful signal, (θu, ϕu), is known, and it is possible to observe the sum of the interference and noise in the absence of the useful signal. Therefore Rx=Rib and (θu, ϕu) are each known;        2nd situation: The position of the useful signal is not known but a portion of the emitted signal (pilot sequences, for example) is precisely known. The sum of the useful signal, of the interference and of the noise is then observed. Therefore Rx=Ru+Rib and u[k] is partially known;        3rd situation: The position of the useful signal is known but it is not possible to observe the sum of the interference and of the noise in the absence of the useful signal. The sum of the useful signal, of the interference and of the noise is then observed. Therefore Rx=Ru+Rib and (θu, ϕu) are each known.        
The first two situations allow a performance that is satisfactory in terms of interference suppression to be obtained using conventional well-known methods, even when amplitude/phase errors are present in the directional vectors of the useful signal and of the interfering signals or when the position (θu, ϕu) of the useful signal is not precisely known. For example, the minimum variance distortionless response (MVDR) method may be used in the first situation and the minimum mean square error (MMSE) method may be implemented in the second situation (H. L. Van Trees, “Optimum Array Processing”, Part IV of Detection, Estimation and Modulation Theory, Wiley Interscience, 2002).
However, in space or terrestrial radio-communications, it is in general not possible (for reasons of spectral efficiency, throughput and system-level constraints) to reserve a time or frequency slot to estimate the correlation matrix in the absence of useful signal. Situation 1 is therefore not possible.
Nevertheless sequences incorporated into the useful signal may be known beforehand, this corresponding to situation 2. However, the suppression of co-channel interference then requires the pilot sequence associated with each user to be identified, this in certain systems notably complicating implementation.
As regards situation 3, the Capon method is the interference suppression method conventionally used, this method consisting in minimizing the total power of the received signal under a constraint of unit directivity in the direction of the useful signal. The weighting w is then calculated in the following way (H. L. Van Trees, “Optimum Array Processing”, Part IV of Detection, Estimation and Modulation Theory, Wiley Interscience, 2002):
                              w          _                =                                            R              x                              -                1                                      ⁢                                          a                _                            ⁡                              (                                                      θ                    u                                    ,                                      ϕ                    u                                                  )                                                                                                          a                  _                                ⁡                                  (                                                            θ                      u                                        ,                                          ϕ                      u                                                        )                                            H                        ⁢                          R              x                              -                1                                      ⁢                                          a                _                            ⁡                              (                                                      θ                    u                                    ,                                      ϕ                    u                                                  )                                                                        (        5        )            
It is only in the absence of amplitude/phase errors and if the position of the useful signal is perfectly known that the conventional Capon method allows a satisfactory performance to be obtained. Specifically, in the presence of errors in the directional vectors or indeed when the position of the useful signal is not precisely known, the Capon method leads to an undesired suppression of the useful signal (FIGS. 2a and 2b). This effect is disadvantageous and manifests itself even when the errors are of low level. The conventional Capon method is therefore not robust with respect to errors when a useful signal is present. The article by Y. Wang et al., “Robust mainlobe interference suppression for coherent interference environment” published in “EURASIP Journal on advances in signal processing” in 2016 gives an example of use of the Capon method to suppress interference signals in the mainlobe of an array of omnidirectional antennas.
FIGS. 2a and 2b show an example of interference suppression for an equispaced linear array (ELA) of 10 elements. In FIG. 2a the useful signal is not suppressed whereas in FIG. 2b the useful signal is suppressed because of a 0.1° error in the position of the useful signal.
In the literature, many methods have been proposed for solving the problem of the robustness of adaptive beamforming in the presence of a useful signal. For the most part these methods may be grouped into the following categories (J. Li and P. Stoica, “Robust adaptive beamforming”, Wiley, 2006; S. A. Vorobyov, “Principles of minimum variance robust adaptive beamforming design”, Signal Processing, 93:3264-3277, 2013):
Diagonal Loading Methods
(B. D. Carlson, “Covariance matrix estimation errors and diagonal loading in adaptive arrays”, IEEE Transactions on Aerospace and Electronic Systems, 24:397-401, 1988): A positive constant λ is added to the diagonal of the correlation matrix Rx of the received signals, which becomes Rx+λIN, where IN is an identity matrix of order N. This allows the addition of a fictional noise masking the useful signal and preventing its suppression to be simulated, λ being the power of the fictional noise. However, the performance of this method is very sensitive to the setting of the constant λ and, at the present time, this problem has not been satisfactorily solved.
Methods with Directivity Constraints, Derivative Constraints, Etc.
(M. H. Er, “Adaptive antenna array under directional and spatial derivative constraints”, IEEE Proceedings H-Microwaves Antennas and Propagation, 135:414-419, 1988): It is a question of attempting to protect the useful signal by applying constraints, so that the directivity of the useful signal remains high enough in the presence of errors. The applied constraints consume degrees of freedom of the antenna, this resulting in a decrease in the capacity to reject interference and therefore in a performance that is not very good.
Methods Based on Orthogonality Between the Signal Subspace and the Noise Subspace
(A. Haimovich et al., “Adaptive antenna arrays using eigenvector methods”, IEEE International Symposium on Antennas and Propagation, 3:980-983, 1988): These methods aim to decrease the effect of errors by virtue of an orthogonal projection. Nevertheless, they are ineffective because the presence of the useful signal is not sufficiently removed.
Methods Based on a Sphere or an Ellipsoid of Uncertainty
(J. Li et al., “On robust Capon beamforming and diagonal loading”, IEEE Transactions on Signal Processing, 51:1702-1715, 2003): They assume, for example, that the discrepancy between the current directional vector and the assumed directional vector of the useful signal is limited, i.e. that ∥acurrent−aassumed∥<ε. The weighting vector used in the beamforming is then computed using an optimization under a constraint of inclusion of the directional vector in the sphere or ellipsoid of uncertainty. These methods give good results provided that the power of the useful signal is lower than the power of the interference. When the useful signal level becomes comparable to or higher than the level of the interference, the useful signal is suppressed. Some of these methods are very complex and expensive in computational resources.
Methods Based on a Reconstruction of the Correlation Matrix of the Interference and of the Noise
(Y. Gu et al., “Robust adaptive beamforming based on interference covariance matrix reconstruction and steering vector estimation”, IEEE Transactions on Signal Processing, 60:3881-3885, 2012): They estimate the directional vectors of the interference so as to construct a matrix {circumflex over (R)}ib that is not greatly different from Rib, so as to return to situation 1 described above, in which the interference suppression methods are robust.
Methods Based on the Estimation of the Directional Vectors
(A. Khabbazibasmen et al., “Robust adaptive beamforming based on steering vector estimation with as little as possible prior information”, IEEE Transactions on Signal Processing, 2974-2987, 2012): They are based on the fact that knowledge of the directional vector of the useful signal is important if the robustness of the Capon method is to be improved. They are not robust when the signal-to-noise ratio of the useful signal is high.
Probabilistic Methods
(S. Vorobyov et al., “On the relationship between robust minimum variance beamformers with probabilistic and worst-case distortionless response constraints”, IEEE Transactions on Signal Processing, 56:5719-5724, 2008): They are based on the observation that position errors and amplitude/phase errors are random, and compute a robust weighting in such a way that the probability of suppression of the useful signal remains low. Just like the preceding methods, these methods have a mediocre performance when the signal-to-noise ratio of the useful signal is high.
Other Methods:
They are mixtures of the preceding methods.